Use index notation to write squares such as 2², 3², 4².
Recognise: •squares of numbers
Use vocabulary from previous years and extend to:
index, indices, index notation, index law…
Use index notation for small integer powers.
For example:
144
= 38486
Know that x0
1, for all values of x. =
Know that: 10–1
1⁄10 = 0.1 10–2 = 1⁄100 = 0.01 =
Know how to use the xy key on a calculator to calculate
powers.
Use ICT to estimate square roots or cube roots to the
required number of decimal places. For example:
• Estimate the solution of x
70. =
The positive value of x lies between 8 and 9, since 82
64 and 92 = 81. =
Try numbers from 8.1 to 8.9 to find a first
approximation lying between 8.3 and 8.4.
Next try numbers from 8.30 to 8.40.
Investigate problems such as:
•Estimate the cube root of 20. •The outside of a cube made from smaller cubes is painted blue.
How many small cubes have 0, 1, 2 or 3 faces painted blue?
Investigate. •Three integers, each less than 100, fit the equation
a2 + b2
c2. =
What could the integers be?
Recognise that:
•
indices are added when multiplying, e.g.
43 × 42
(4 × 4 × 4) × (4 × 4) =
4 × 4 × 4 × 4 × 4 =
45 = 4(3 + 2) =
•indices are subtracted when dividing, e.g.
45 ÷ 42
(4 × 4 × 4 × 4 × 4) ÷ (4 × 4) =
4 × 4 × 4 =
43 = 4(5 – 2) =
•42 ÷ 45
4(2 – 5) = 4–3 =
•75 ÷ 75
70 = 1 =
Generalise to algebra. Apply simple instances of the
index laws (small integral powers), as in: •n2 × n3
n2 + 3 = n5 •p3 ÷p2 = p3 – 2 = p =
Squaring a number 32 means ‘3 squared’, or 3 x 3.
The small 2 is an index number, or power. It tells us how many times we should multiply 3 by itself.
Similarly 72 means ‘7 squared’, or 7 x 7.
And 102 means ‘10 squared’, or 10 x 10.
So, 12
1 x 1
1
22
2 x 2
4
32
3 x 3
9
Etc.
1, 4, 9, 16, 25… are known as square numbers. Square roots The opposite of a square number is a square root.
We use the symbol to mean square root.
So we can say that = 2 and = 5.
However, this is not the whole story, because -2 x -2 is also 4, and -5 x -5 is also 25.
So, in fact, = 2 or -2. And = 5 or -5.
Remember that every positive number has two square roots. Cubing a number 2 x 2 x 2 means ‘2 cubed’, and is written as 23.
13
1 x 1 x 1
1
23
2 x 2 x 2
8
33
3 x 3 x 3
27
Etc
1, 8, 27, 64, 125… are known as cube numbers. Cube roots The opposite of a cube number is a cube root. We use the symbol to mean cube root.
So is 2 and is 3.
Each number only has one cube root.
Index notation Index notation is used to represent powers, for example
a2 means a × a and here the index is 2
b3 means b × b × b and here the index is 3
c4 means c × c × c × c and here the index is 4
etc.
When there is a number in front of the variable:
4d2 means 4 × d × d.
2e3 means 2 × e × e × e
Index laws Multiplying and dividing When multiplying you add the indices, and when dividing you subtract the indices. So it follows that:
p3 × p7
p
10, and s5 ÷ s3
s2
For the expression:
4s3 x 3s2
The numbers in front of the variables follow the usual rules of multiplication and division, but index numbers follow the rules of indices. So we multiply 4 and 3 and add 3 and 2
4s3 × 3s2 = 12s5 Question What is 3c2 × 5c4?
Answer To work it out:
Add the indices:
2 + 4 = 6
Multiply the numbers in front of the variable:
3 x 2
Answer:
3c2 × 5c4 = 15c6
Take care when multiplying and dividing expressions such as y × y4 or z3 ÷ z.
y is the same as y1, so y × y4 = y5.
z is the same as z1, so z3 ÷ z = z2.
Adding and subtracting You can only add and subtract ‘like terms’. 3, 4 and 20 are all like terms (because they are all numbers).
a, 3a and 200a are all like terms (because they are all multiples of a).
a2, 10a2 and -2a2 are all like terms (because they are all multiples of a2)
You cannot simplify an expression like 4p + p2 because 4p and p2 are not like terms.
But you can simplify 3r2 + 5r2 + r2.
3r2 + 5r2 + r2 tells us that we have ‘three lots of r2’ + ‘five lots of r2’ + ‘one lot of r2’ - so in total ‘nine lots of r2’, or 9r2.
So, 3r2 + 5r2 + r 2 = 9r2 Question What is s2 + 8s2 - 2s2?
Answer Answer: 7s2
Remember that 1 + 8 - 2 = 7, so s2 + 8s2 - 2s2 = 7s2 Remember that if we have a mix of terms we must gather like terms before we simplify. Example
3p2 + 2p + 4 - 2p2 + 5 = 3p2 - 2p2 + 2p + 4 + 5 = p2 + 2p + 9
Substitution You might be asked to substitue a number into an expression.
For example , what is the value of 4p3 when p = 2?
We know that 4p3 means 4 × p × p × p, so when p = 2 we substitute this into the expression: 4 × 2 × 2 × 2 (or 4 × 23) = 32 Question What is the value of 4y2 - y, when y = 3?
Using a calculator A calculator can be used to work out one number to the power of another.
The index button is usually marked xy or yx.
Sometimes you need to press SHIFT or 2ndFto use this button
For example, to calculate 54, you may need to press
You should find out which buttons you need to use on your calculator.
Make sure that you get the correct answer of 625 for the calculation above.
TEST NameDate 1. s² means:
s x s
s + s
2s
2. 2w² means:
2w x 2w
2 x w x w
4w
3. y7 x y2 is the same as:
y5
y9
y14
4. y10 ÷ y2 is the same as:
y5
y8
y20
5. Simplify 4w6 x 2w2
6w8
8w12
8w8
6. Simplify 14h8 ÷ 2h2
7h4
7h6
12h6
7. Simplify p x 2p2 x 5p3
7p5
10p5
10p6
8. What's the value of 4k2 when k = 5?
20
100
400
9. What's the value of 2j2 + j when j = 3?
21
39
400
10. What's the value of 2p2 + p3 - p when p = 2?
14
22
40
1. Find some triples of whole numbers a, b and csuch that a2+b2+c2is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
2. What can you say about the values of n that make
7 n + 3 n a multiple of 10?
Are there other pairs of integers between 1 and 10 which have similar properties?
Use index notation to write squares such as 2², 3², 4².
Recognise: • squares of numbers
Use vocabulary from previous years and extend to:
index, indices, index notation, index law…
Use index notation for small integer powers.
For example:
144= 38486
Know that x0
1, for all values of x. =Know that: 10–1
1⁄10 = 0.1 10–2 = 1⁄100 = 0.01 =Know how to use the xy key on a calculator to calculate
powers.
Use ICT to estimate square roots or cube roots to the
required number of decimal places. For example:
• Estimate the solution of x
70. =The positive value of x lies between 8 and 9, since 82
64 and 92 = 81. =Try numbers from 8.1 to 8.9 to find a first
approximation lying between 8.3 and 8.4.
Next try numbers from 8.30 to 8.40.
Investigate problems such as:
• Estimate the cube root of 20. • The outside of a cube made from smaller cubes is painted blue.
How many small cubes have 0, 1, 2 or 3 faces painted blue?
Investigate. • Three integers, each less than 100, fit the equation
a2 + b2
c2. =What could the integers be?
Recognise that:
•
indices are added when multiplying, e.g.
43 × 42
(4 × 4 × 4) × (4 × 4) =
4 × 4 × 4 × 4 × 4 =
45 = 4(3 + 2) =• indices are subtracted when dividing, e.g.
45 ÷ 42
(4 × 4 × 4 × 4 × 4) ÷ (4 × 4) =
4 × 4 × 4 =
43 = 4(5 – 2) =• 42 ÷ 45
4(2 – 5) = 4–3 =• 75 ÷ 75
70 = 1 =Generalise to algebra. Apply simple instances of the
index laws (small integral powers), as in: • n2 × n3
n2 + 3 = n5 • p3 ÷ p2 = p3 – 2 = p =Squaring a number
32 means ‘3 squared’, or 3 x 3.
The small 2 is an index number, or power. It tells us how many times we should multiply 3 by itself.
Similarly 72 means ‘7 squared’, or 7 x 7.
And 102 means ‘10 squared’, or 10 x 10.
So, 12
1 x 1
122
2 x 2
432
3 x 3
9Etc.
1, 4, 9, 16, 25… are known as square numbers.
Square roots
The opposite of a square number is a square root.
We use the symbol to mean square root.
So we can say that = 2 and = 5.
However, this is not the whole story, because -2 x -2 is also 4, and -5 x -5 is also 25.
So, in fact, = 2 or -2. And = 5 or -5.
Remember that every positive number has two square roots.
Cubing a number
2 x 2 x 2 means ‘2 cubed’, and is written as 23.
13
1 x 1 x 1
123
2 x 2 x 2
833
3 x 3 x 3
27Etc
1, 8, 27, 64, 125… are known as cube numbers.
Cube roots
The opposite of a cube number is a cube root. We use the symbol to mean cube root.
So is 2 and is 3.
Each number only has one cube root.
Index notation
Index notation is used to represent powers, for example
a2 means a × a and here the index is 2
b3 means b × b × b and here the index is 3
c4 means c × c × c × c and here the index is 4
etc.
When there is a number in front of the variable:
4d2 means 4 × d × d.
2e3 means 2 × e × e × e
Index laws
Multiplying and dividing
When multiplying you add the indices, and when dividing you subtract the indices.
So it follows that:
p3 × p7
p
10, and s5 ÷ s3
s2For the expression:
4s3 x 3s2
The numbers in front of the variables follow the usual rules of multiplication and division, but index numbers follow the rules of indices. So we multiply 4 and 3 and add 3 and 2
4s3 × 3s2 = 12s5
Question
What is 3c2 × 5c4?
Answer
To work it out:
- Add the indices:
- 2 + 4 = 6
- Multiply the numbers in front of the variable:
- 3 x 2
- Answer:
- 3c2 × 5c4 = 15c6
Take care when multiplying and dividing expressions such as y × y4 or z3 ÷ z.y is the same as y1, so y × y4 = y5.
z is the same as z1, so z3 ÷ z = z2.
Adding and subtracting
You can only add and subtract ‘like terms’.
3, 4 and 20 are all like terms (because they are all numbers).
a, 3a and 200a are all like terms (because they are all multiples of a).
a2, 10a2 and -2a2 are all like terms (because they are all multiples of a2)
You cannot simplify an expression like 4p + p2 because 4p and p2 are not like terms.
But you can simplify 3r2 + 5r2 + r2.
3r2 + 5r2 + r2 tells us that we have ‘three lots of r2’ + ‘five lots of r2’ + ‘one lot of r2’ - so in total ‘nine lots of r2’, or 9r2.
So, 3r2 + 5r2 + r 2 = 9r2
Question
What is s2 + 8s2 - 2s2?
Answer
Answer: 7s2
Remember that 1 + 8 - 2 = 7, so s2 + 8s2 - 2s2 = 7s2
Remember that if we have a mix of terms we must gather like terms before we simplify.
Example
3p2 + 2p + 4 - 2p2 + 5 = 3p2 - 2p2 + 2p + 4 + 5 = p2 + 2p + 9
Substitution
You might be asked to substitue a number into an expression.
For example , what is the value of 4p3 when p = 2?
We know that 4p3 means 4 × p × p × p, so when p = 2 we substitute this into the expression: 4 × 2 × 2 × 2 (or 4 × 23) = 32
Question
What is the value of 4y2 - y, when y = 3?
Using a calculator
A calculator can be used to work out one number to the power of another.
The index button is usually marked xy or yx.
Sometimes you need to press SHIFT or 2ndFto use this button
For example, to calculate 54, you may need to press
You should find out which buttons you need to use on your calculator.
Make sure that you get the correct answer of 625 for the calculation above.
TEST
Name Date
1. s² means:
s x s
s + s
2s
2. 2w² means:
2w x 2w
2 x w x w
4w
3. y7 x y2 is the same as:
y5
y9
y14
4. y10 ÷ y2 is the same as:
y5
y8
y20
5. Simplify 4w6 x 2w2
6w8
8w12
8w8
6. Simplify 14h8 ÷ 2h2
7h4
7h6
12h6
7. Simplify p x 2p2 x 5p3
7p5
10p5
10p6
8. What's the value of 4k2 when k = 5?
20
100
400
9. What's the value of 2j2 + j when j = 3?
21
39
400
10. What's the value of 2p2 + p3 - p when p = 2?
14
22
40
1. Find some triples of whole numbers a, b and c such that a2+b2+c2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
2. What can you say about the values of n that make
7 n + 3 n a multiple of 10?
Are there other pairs of integers between 1 and 10 which have similar properties?